I added a light to an exsisting ceiling that has four recess lights. The new light was connected to a wires that had a pull chain light which the four recess lights are connected too. Adding the new light in the fixture where the pull chain light was, now the light will not turn off. But the recess lighting turns on/off normally. The wiring is three white wires connected and three black wires connected. I connected the new light following the same format black to black and white to white. Again the recess lighting works as it did before but the new light remains on. At the switch there is two white wires connected and the two black wires are going to the single pole switch.
What's going on here!

Are you sure the old ceiling light was on the switch circuit? Usually a pull chain fixture is not wired through a wall switch, it is wired from an outlet and the pull chain intercepts the power. If you put a VOM or test light connected to the old ceiling fixture wiring. I bet the switch doesn't control the circuit. You will need to access the wiring from the cans and pull a lead from them to the new fixture.